Problem 5 - Fundamentals of Mathematics

Question

Translate each phrase or sentence into a mathematical expression or equation. Two ninths of a number is eleven.

Step-by-Step Solution

Answer

The equation is \( \frac{2}{9}x = 11 \).

Step 1: Identify the unknown

The problem mentions 'a number'. This is our unknown, and we will represent it by a variable, commonly used is \( x \).

Step 2: Translate 'two ninths of a number' into an expression

The phrase 'two ninths of a number' can be translated into the mathematical expression \( \frac{2}{9} \times x \). This means two ninths times the unknown number.

Step 3: Set up the equation based on the sentence

The full sentence is 'Two ninths of a number is eleven.' The word 'is' translates to an equals sign in mathematical language, so we set up the equation \( \frac{2}{9} \times x = 11 \).

Key Concepts

Translating Phrases into Equations

In mathematics, translating verbal phrases into mathematical equations is a crucial skill. It allows us to express real-world problems as solvable mathematical problems. Let's take the phrase "two ninths of a number is eleven".
This phrase involves both a fraction "two ninths" and a statement of equality "is eleven". Here is how you break it down:

• "Two ninths of a number" is translated into a fraction times an unknown variable, commonly written as \( \frac{2}{9} \cdot x \), where \( x \) is the unknown number.
• "Is eleven" is translated into an equation where the expression is set equal to 11, written as \( \frac{2}{9} \cdot x = 11 \).

This method of translation involves identifying important parts of the phrase: numbers, operations, and relationships, and expressing them in mathematical language using symbols.

Unknown Variable Representation

Another key concept in forming equations is the representation of unknown values using variables. When a problem mentions an unknown quantity, like "a number" in the exercise, it is often represented by a letter, such as \( x \).
Using a variable:

• Gives us a way to write algebraic expressions even when we don't know the exact value of the quantity it represents.
• Makes it possible to perform operations on quantities that are unknown or variable.

In this context, it's important to remain consistent. If \( x \) represents the unknown in your equation, continue using \( x \) throughout the problem to avoid confusion.

Setting Up Equations

Setting up equations is the process of creating a mathematical statement that connects an unknown variable to known values. Using our exercise as an example, this involves both understanding the problem and translating it accurately into an equation.
To set up an equation:

• Identify the component of the phrase that indicates an operation (e.g., multiplication, addition).
• Use an equality sign where the phrase indicates a relationship like "is", "are" or "equals".
• Check the equation by ensuring all parts of the verbal statement have corresponding symbols in the equation.

In the example, the phrase "two ninths of a number is eleven" becomes \( \frac{2}{9} \cdot x = 11 \). This equation accurately represents the relationship described in the sentence and is ready to be solved to find the value of \( x \).